\(KK\)-theory of A-valued semi-circular systems
type: Item
instance of: scholarly article
title: \(KK\)-theory of A-valued semi-circular systems
zbMATH Open document ID: 1339.19001
author: Emmanuel Germain
author: Pierre Umber
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2016-05-12T00:00:00Z
reviewed by: Ralf Meyer
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 6579852
zbMATH Keywords: Toeplitz \(C^*\)-algebra
zbMATH Keywords: \(C^*\)-correspondence
zbMATH Keywords: \(KK\)-equivalence
MaRDI profile type: Publication
full work available at URL: j.crma.2015.10.013
OpenAlex ID: W2196156721
cites work: Q4335109
cites work: Free quasi-free states
cites work: \(A\)-valued semicircular systems
cites work: The \(K\)-groups of the \(C^*\)-algebra of a semicircular family
cites work: Free Random Variables
DOI: 10.1016/J.CRMA.2015.10.013
review text: Let \(E\) be a Hilbert \(A\)-module with a \(^*\)-homomorphism from \(A\) to the \(C^*\)-algebra of adjointable operators on \(E\). A representation of \(E\) on a \(C^*\)-algebra \(B\) is a pair consisting of a \(^*\)-homomorphism \(f\) from \(A\) to and a linear map \(S\) from \(E\) to \(B\) with \(S(a_1 x a_2) = f(a_1) S(x) f(a_2)\) for all \(a_1,a_2\in A\), \(x\in E\) and \(S(x_1)^* S(x_2) = f(\langle x_1,x_2\rangle)\) for all \(x_1,x_2\in E\). The Toeplitz \(C^*\)-algebra \(T_E\) is the universal \(C^*\)-algebra for such representations.NEWLINENEWLINE\textit{M. V. Pimsner} [Fields Inst. Commun. 12, 189--212 (1997; Zbl 0871.46028)] has shown that the canonical inclusion of \(A\) into \(T_E\) is a \(KK\)-equivalence. Here this is generalised as follows: the inclusion of \(A\) into \(S\) is a \(KK\)-equivalence, where \(S\) is a \(C^*\)-subalgebra of \(T_E\) generated by \(A\) and some set of linear combinations of creation and annihilation operators. In particular, if \(E\) carries an involution, \(S\) could be the \(C^*\)-subalgebra generated by \(A\) and elements of the form \(T_x + T_{x^*}^*\) for \(x\in E\), where \(T_x\) denotes the element of \(T_E\) associated to \(x\in E\). This contains a \(K\)-theory computation by \textit{D. Voiculescu} [K-Theory 7, No. 1, 5--7 (1993; Zbl 0786.46054)] for a semicircular family as a special case.
description: scientific article; zbMATH DE number 6579852